NIPHER PROPERTY OF THE ISENTROPIC CURVE. 409 



a= _ ;, = - ^ = tan/. . . (9) 



For very low pressures, the direction of maximum slope -^ 

 becomes more and more nearly at right angles to the plane of 

 _/J, V ; while for high pressures this direction becomes more and 

 more nearly parallel to the plane of p, v. The direction of maxi- 

 mum slope is constant along a line of constant pressure. 



2°. To find the direction of the isentropic line at any point on 

 the surface, as related to the direction of maximum slope deter- 

 mined in (9). 



Poisson's equation : 



T^-y*"^ nr const. . . . (lo) 



is a projection of the isentropic line upon the plane of v T, k 

 being the ratio of the specific heats = 1.4 j. 



Calling a' the tangent of the angle which any element of this 

 projection makes with the v axis, we have 



a' := — = tan i' . 

 dv 



This value of a' is obtained by differentiating (10) and is found 



to be 



^-I = _ ^(^_i)=_^(^_i). . (II) 

 dv V rC 



Here also the condition of constant pressure gives a constant 

 value for a' . Hence, at any point along any line of constant press- 

 ure the projection of an element of the isentropic line, upon the 

 -J, T plane, makes a constant angle with the projected line of 

 greatest slope at the same point. 



From equations (9) and (11) it follows that 



tan /' = ^ — . ; . . . . (12) 



tan / 



from which it will appear that for either very high or very low 

 pressure the isentropic line runs at right angles to the direction 

 of greatest slope. The condition that it shall coincide with the 

 direction of greatest slope is 



\R 



tan z' = ^^_i — -—, 



